# Newton vs Relativity ...

**URL:** <https://boards.straightdope.com/t/newton-vs-relativity/38540>\
**Category:** Factual Questions\
**Created:** [October 27, 2000, 3:18am UTC](https://boards.straightdope.com/t/newton-vs-relativity/38540 "2000-10-27T03:18:59Z")\
**Posts on this page:** 3\
**Page:** 1

<div class="post-metadata">

**Author:** ![DeutschFox](https://avatars.discourse-cdn.com/v4/letter/d/49beb7/32.png) [@DeutschFox](https://boards.straightdope.com/u/DeutschFox)\
**Post date:** [October 27, 2000, 3:18am UTC](https://boards.straightdope.com/t/newton-vs-relativity/38540/1 "2000-10-27T03:18:59Z")

</div>

I understand that it can be shown that kinetic energy at speeds v\<\<c can be derived from the relativistic formula E=mc^2 to show the Newtonian KE=(1/2)mv^2. Hmmmm?

---

<div class="post-metadata">

**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\
**Post date:** [October 27, 2000, 3:29am UTC](https://boards.straightdope.com/t/newton-vs-relativity/38540/2 "2000-10-27T03:29:55Z")

</div>

Well, yes, but you need another formula. The e=mc[sup]2[/sup] includes both kinetic energy and rest mass energy, because m is relativistic mass which gets bigger as the speed increases. The kinetic energy is ke=mc[sup]2[/sup] - m[sub]0[/sub]c[sup]2[/sup] where m[sub]0[/sub] is the rest mass of the particle. Relativistic mass is m[sub]0[/sub]/sqrt(1-v[sup]2[/sup]/c[sup]2[/sup]). Plug that into the ke formula and I _think_ you can simplify it down to ke=1/2 m[sub]0[/sub] v[sup]2[/sup], if you take the limit as v/c \<\< 1.

---

<div class="post-metadata">

**Author:** ![Opus1](https://avatars.discourse-cdn.com/v4/letter/o/858c86/32.png) [@Opus1](https://boards.straightdope.com/u/Opus1)\
**Post date:** [October 27, 2000, 4:51am UTC](https://boards.straightdope.com/t/newton-vs-relativity/38540/3 "2000-10-27T04:51:06Z")

</div>

You need to use a Taylor’s expansion to reduce it further. Sqrt[1-x]~= 1-(x/2) for very small x. In this particular case, x = v^2/c^2. Substitute 1-(2v^2/c^2) for Sqrt[1-v^2/c^2] and you’ll reduce it down.